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I would suggest that you repair your phone with update service
make sure to backup everything before hand
Update Service (SEUS)
http://dl-www.sonyericsson.com/cws/download/1/110/61/1317636927/Update_Service_Setup-2.11.10.7.exe
1st go to
settings -> about phone -> android version
do you have 2.3.4? if you do go to
settings -> privacy -> and make sure both options are selected
back up my data and automatic restore
if you do NOT have 2.3.4 then use this methods to backup your phone
Backup & Restore app
Use this links to download the apk
http://forum.xda-developers.com/showthread.php?t=1193802
or
http://www.elita-clan.com/downloads/backup_and_restore_v010935.apkhttp://www.elita-clan.com/downloads/backup_and_restore_v010935.apkor
http://www.mediafire.com/?1n07llajeaoo03y
How to install it
You need to
go to settings -> applications -> check unknown sources
download IO File Manager
https://market.android.com/details?id=org.openintents.filemanager&feature=search_result
download the backup & restore app, transfer it to your SD card, use OI file manager to find it -> tap on it -> install it
Contacts
just to play it safe sync your contacts w/ google if you haven't already
SMS/MMS
if you don't want to use the backup & restore app then
Install "backup SMS" to back up your SMS and MMS to your Gmail
https://market.android.com/details?id=com.zegoggles.smssync&feature=search_result
or your SMS use
"backup & restore SMS"
https://market.android.com/details?id=com.riteshsahu.SMSBackupRestore&feature=search_result
or for your MMS use "Save MMS"
https://market.android.com/details?id=com.schwimmer.android.mmsextract&feature=search_result
Call Logs
And for call logs this is great
https://market.android.com/details?id=com.yang.android.ansta
This is very low..
For how much time the music player was on..??
1 hour and an half
how long was the screen on?
Oh but I see 3g sign , so that means yiur mobile metwork data was on for full 8 hours right..??
Was auto sync on..??
Fo to settings > accounts and sync..and check..
If it was on and yoi were also surfing the net from time to time then it can be normal because 3g consumes more batery..
Just listem to music with 3g off..and yoir bttery life will get double I think so 
To save battery you should disabke 3g conne tion when not in use..
If you can , just check the battery life with 3g off..ok.and then post your result 
Also..i see a icon of android..with 13 number on it..whats that..?? Task killer..?? Uninstall it right away if its the task killer..no need for task killer now..it rather harms the battery..ok
13 is battery widget, the screen has the brightness OFF and autosync is OFF.
i used in 8 hours data active, but i don't think that ONLY 3g is the cause of 8 hour lasting..the phone is new!! i bought it 5 days ago!!
I'm currently using 3g to reply here.. and I used 3g for 30 minutes with brightnes 15% and battery dropped to 53% from 66% ;( in just 20-25 minutes of using 3g..
Is your accounts sync on.?
Go to settings > display and turn off the bravia engine..
Then charge your phone to a full 100% , use it as you normalyy do..if possible turn off the 3g when not in use and make sure yoir bravia engine is off..just do it..afyer the battery reaches around 15% . Post the result here ok.. 
the 3G shouldn't be the cause of the problem, now, as slimshaddy said, do you have a lot of accounts syncing all the time?
like facebook is it updating all the time, every hour, or manually?
are you sure the brightness was off? (0%)
you can download this from the market
Power Control
facebook is every hour.. can't be simply a problem with the battery??
there's one way to find out if it's a problem with the battery,
1st Charge your phone 100%
2nd turn off/on your phone
3rd charge it a bit more just to make sure it's at 100%
4th go to
settings -> accounts & sync -> uncheck Actitivate auto sync
5th use it as you normally will for 8hrs 20minutes and take the screenshots
but do NOT turn on auto sync at all